On the Sample Complexity and Optimization Landscape for Quadratic Feasibility Problems

P Thaker
June 2020
ISIT 2020
OptimizationNonconvex OptimizationSample ComplexityPhase RetrievalSignal Processing
arXiv: 2002.01066

Abstract

We consider the problem of recovering a complex vector from quadratic measurements, known as quadratic feasibility, which encompasses the well known phase retrieval problem and has applications in power system state estimation and x-ray crystallography. While the quadratic feasibility problem is generally NP-hard and may be unidentifiable, we establish conditions under which this problem becomes identifiable, particularly when the matrices are Hermitian matrices sampled from a complex Gaussian distribution. We explore a nonconvex optimization formulation and establish features of the optimization landscape that enables gradient algorithms with arbitrary initialization to converge to a globally optimal point with high probability. Our results also reveal sample complexity requirements for successfully identifying a feasible solution.

What Excited Me

This paper tackles one of the most fundamental challenges in computational mathematics: recovering complex signals from nonlinear measurements. What excites me most is the elegant theoretical analysis that transforms a seemingly intractable NP-hard problem into something we can actually solve with guarantees! The beauty lies in how the optimization landscape analysis reveals hidden geometric structures that make gradient descent work despite the nonconvex nature of the problem. The connection between sample complexity and optimization landscape is particularly fascinating - it's like discovering that the hardness of a problem isn't just about the algorithm you choose, but about having the right amount of data to make the landscape 'nice' enough for simple algorithms to succeed. The applications spanning from power systems to crystallography show how fundamental mathematical insights can have broad real-world impact.

Problem Formulation

The quadratic feasibility problem seeks to recover a complex vector x from a collection of quadratic measurements of the form:

{⟨Aᵢx, x⟩}ᵢ₌₁ᵐ

This problem generalizes the famous phase retrieval problem and has significant applications across multiple domains.

Key Applications

1. Phase Retrieval

  • Fundamental problem in signal processing and imaging
  • Recovery of signals from magnitude-only measurements
  • Critical in optical imaging and astronomical observations

2. Power System State Estimation

  • Monitoring and control of electrical power grids
  • Real-time estimation of system states from measurements
  • Essential for grid stability and optimization

3. X-ray Crystallography

  • Determining molecular structures from diffraction patterns
  • Recovering phase information lost in measurement process
  • Critical for drug discovery and materials science

Theoretical Contributions

1. Identifiability Conditions

  • Challenge: Quadratic feasibility is generally NP-hard and may be unidentifiable
  • Solution: Established precise conditions under which the problem becomes identifiable
  • Key Insight: Hermitian matrices sampled from complex Gaussian distributions provide favorable properties

2. Optimization Landscape Analysis

  • Nonconvex Formulation: Despite nonconvexity, the optimization landscape has special structure
  • Global Convergence: Proved that gradient algorithms with arbitrary initialization converge to globally optimal points with high probability
  • Geometric Properties: Characterized features of the optimization landscape that enable tractable solutions

3. Sample Complexity Bounds

  • Data Requirements: Established minimum number of measurements needed for successful recovery
  • Probabilistic Guarantees: Provided high-probability bounds on algorithm performance
  • Practical Implications: Revealed the trade-off between measurement complexity and recovery guarantees

Technical Innovation

Isometry Properties

The paper proves crucial isometry properties when measurement matrices are:

  • Hermitian matrices sampled from complex Gaussian distributions
  • This structure enables tractable analysis despite the nonconvex nature of the problem

Optimization Landscape Characterization

Key findings about the optimization landscape:

  • No Spurious Local Minima: Under certain conditions, all local minima are global minima
  • Favorable Geometry: Gradient descent can escape saddle points and converge to global solutions
  • Arbitrary Initialization: No need for careful initialization strategies

Theoretical Significance

This work bridges several important areas:

  • Computational Complexity: Shows how statistical assumptions can make NP-hard problems tractable
  • Optimization Theory: Demonstrates that nonconvex optimization can have benign landscapes
  • Statistical Learning: Connects sample complexity with optimization difficulty

Impact on Nonconvex Optimization

The paper contributes to the growing understanding that many practically important nonconvex optimization problems have favorable geometric properties that enable efficient algorithms. This challenges the traditional view that nonconvexity necessarily implies computational intractability.

Algorithmic Implications

The theoretical guarantees suggest that simple gradient-based methods can be highly effective for quadratic feasibility problems, provided:

  1. Sufficient measurements are available (sample complexity bounds)
  2. Measurement matrices have appropriate statistical properties
  3. Problem parameters satisfy identifiability conditions

This work exemplifies how rigorous theoretical analysis can provide both practical algorithms and fundamental insights into the nature of computational complexity in signal recovery problems.